Phase Estimation from Atom Position Measurements

Abstract

We study the measurement of the position of atoms as a means to estimate the relative phase between two Bose-Einstein condensates. First, we consider N atoms released
from a double-well trap, forming an interference pattern, and show that a simple least-squares fit to the density gives a shot-noise limited sensitivity. The shot-noise limit can instead be overcome by using correlation functions of order √N or larger. The measurement of the Nth-order correlation function allows to estimate the relative phase at the Heisenberg limit.
Phase estimation through the measurement of the center-of-mass of the interference
pattern can also provide sub-shot-noise sensitivity.
Finally, we study the effect of the overlap between the two clouds on the phase estimation, when Mach-Zehnder interferometry is performed in a double-well.
We find that a non-zero overlap between the clouds dramatically reduces the phase sensitivity.

pacs:

07.60.Ly, 03.75.Dg, 37.25.+k, 67.85.-d

1 Introduction

Interferometry aims at the estimation of the relative phase between two
wave-packets. In a standard optical interferometer, like the well known Mach-Zehnder setup [1], these two wave-packets correspond to the light traveling
inside the two arms of the device, and the relative phase θ is acquired, for instance, as a result of different optical path length.
After the phase is accumulated, the two wave-packets are recombined through a beam-splitter, and the signal at the two output ports depends on θ.
The phase can be estimated by measuring the difference in intensities between these ports.
Apart from photons, atoms can be employed for interferometric purposes as well [2]. The atoms present some interesting advantages with respect to light,
especially due to the non-zero mass. In particular, the creation of atomic Bose-Einstein condensates (BECs) opened a new chapter in the field of interferometry.
The BEC, which behaves like a macroscopic matter-wave, constitutes a coherent
and well-controllable source of particles.
This makes the BEC a promising system to measure the electromagnetic [3, 4, 5] or
gravitational [6, 7, 8] forces. Moreover, the
inter-atomic interactions in the BEC are a source of nonlinearity, which can be used to create non-classical states
[9, 10, 11, 12] useful to overcome the limit imposed by the classical physics on measurement precision [13, 14].

A BEC interferometer can be implemented using a double-well trap [15, 16, 17, 18, 19, 20, 21, 22, 23], where the two wave-packets are localized about the two minima of the external potential.
In such configuration, a relative phase θ can be accumulated by letting the system evolve in time in presence of an energy difference between the two potential minima,
and in absence of coupling between the two wells. After this stage, one can, for example, recombine the wave-packets by implementing a beam splitter
(thereby realizing a Mach-Zehnder interferometer).
This will imply a further dynamical evolution during which atoms oscillate between the wells for a time which must be precisely under control, and over which interactions are negligible.

The recombination of the wave-packets can be done in a simpler way, just by releasing them form the double-well trap, so they form an interference pattern, as shown in Fig.1.
In this manuscript we discuss how the information about the phase can be extracted from this pattern
and derive the sensitivity for different estimation strategies.
The manuscript is organized as follows. In Section 2 we formulate the problem and introduce the basic tool –
the N-body correlation function, where N is the total number of atoms.
In Section 3 we demonstrate that by performing a least-squares fit to the measured density [15],
the estimation sensitivity Δθ is bounded by the shot-noise. As discussed in detail in Section 4,
in order to overcome this limit, high-order spatial correlation functions must be measured, namely, of order not smaller than √N.
In particular, when estimation is performed using the N-th order correlation function,
the sensitivity saturates the bound set by the Quantum Fisher Information (QFI) [24]. Then, in Section 5
we analyze an estimation scheme based on the detection of the position of the center-of-mass of the interference pattern,
which can still yield sub-shot-noise sensitivity.
Finally, in Section 6 we study the sensitivity of the
Mach-Zehnder interferometer implemented in a double-well, and we observe that a non-zero overlap between the wave-packets dramatically reduces the sensitivity.
Some details of the calculations are
presented in the Appendix. The present manuscript is an extension of our previous work [25].

2 The model

To begin the discussion of different estimation methods based on position measurement,
we introduce the two-mode field operator of a bosonic gas in a
double-well potential,

^Ψ(x,t)=ψa(x,t)^a+ψb(x,t)^b,

where ^a†/^b† creates
an atom in the left/right well. With the atoms trapped, the relative phase θ is imprinted between
the modes.
This stage is represented by a unitary evolution ^U(θ)=e−iθ^Jz
of the initial state |ψin⟩ of the double-well system.
The three operators

^Jx≡(^a†^b+^b†^a)/2,^Jy≡(^a†^b−^b†^a)/2iand^Jz≡(^a†^a−^b†^b)/2

form a closed algebra of angular momentum. With the phase acquired, the trap is switched off and
the two clouds described by the mode functions ψa/b(x,t) freely expand.

The most general quantity, containing the statistical information about the positions of particles forming the interference pattern, is
the conditional probability of finding N particles at positions →xN=(x1…xN). It can be
expressed in terms of the N-th order correlation
function pN(→xN|θ)=1N!GN(→xN,θ), where

GN(→xN,θ)=⟨ψout|^Ψ†(x1,t)…^Ψ†(xN,t)^Ψ(xN,t)…^Ψ(x1,t)|ψout⟩.

Here, |ψout⟩ denotes the state after the interferometric transformation,
|ψout⟩=e−iθ^Jz|ψin⟩.
To provide a compact and useful expression for this probability, we take following steps.

First, we decompose the initial state in the well-population basis,
|ψin⟩=∑Nn=0Cn|n,N−n⟩ and suppose
that the expansion coefficients are real and posses the symmetry Cn=CN−n. As we will argue later, such choice of Cn’s is natural in context of this work.
We switch from the Schrödinger to the Heisenberg representation, where the field operator evolves according to,

^Ψθ(x,t)≡^U†(θ)^Ψ(x,t)^U(θ)=ψa(x,t)eiθ2^a+ψb(x,t)e−iθ2^b.

The next setp is to introduce the basis of the coherent phase states [26] defined as

|φ,N⟩=1√2NN!(^a†+eiφ^b†)N|0⟩

(where |0⟩ is the state with zero particles). The action of the field operator on these states can be written in a simple form,

^Ψθ(x,t)|φ,N⟩=√N2uθ(x,φ;t)|φ,N−1⟩,

where uθ(x,φ;t)=ψa(x,t)ei2(θ+φ)+ψb(x,t)e−i2(θ+φ).
Next, we expand the Fock states in the basis of the coherent states

|n,N−n⟩=√2N1√(Nn)∫2π0dφ2πe−iφ(N−n)|φ,N⟩.

Thus we can easily write the result of action of the field operator on the input state,

^Ψθ(x,t)|ψin⟩=√2NN2N∑n=0Cn√(Nn)∫2π0dφ2πe−iφ(N−n)uθ(x,φ;t)|φ,N−1⟩.

(1)

Now evaluation of GN(→xN,θ) (and therefore the pN(→xN,θ) as well) is straightforward – we act N times with ^Ψθ on the input state
and calculate the modulus square of the result. After normalization we obtain

pN(→xN|θ)

=

2π∫02π∫0dφ2πdφ′2πN∏i=1u∗θ(xi,φ;t)uθ(xi,φ′;t)

(2)

×

N∑n,m=0CnCmcos[φ(N2−n)]cos[φ′(N2−m)]√(Nn)(Nm).

In the remaining part of the manuscript, we fix t large enough so that the interference
pattern is formed. In this regime, the physical properties of the system change only by scaling ∼√t of the characterisitc dimensions of the system.
The probability (2) is the starting point for the following discussion of various phase estimation strategies.

Figure 1:
Schematic representation of the interferometric procedure. First, a relative phase θ is
imprinted between the wells. Then, the BECs are released from the trap and form an interference pattern. The
detectors (symbolically represented as open squares)
measure the positions of atoms and this data is a starting point for the phase estimation.

3 Estimation via the fit to the density

The simplest way of estimating the value of θ is by fitting
the average density to the interference pattern, as the position of the maximum depends on the relative phase between the two clouds.
Such fit is commonly employed with BECs in double-wells, in order to determine, for instance, the phase coherence in the system [6, 9, 15].

In the experimental realization, the interference pattern is sampled using M bins located at positions xi (i=1…M).
The number of particles ni in each bin
is measured m times, giving the set n(k)i,k=1,...,m. The average occupation ⟨ni⟩=limm→∞∑mk=1n(k)i/m with free parameter θ is then fitted to the histogram
of the measured density {xi,¯ni},i=1,...,M, where ¯ni=∑mk=1n(k)i/m.
If the size Δx of each of M bins is small, ⟨ni⟩
is given by the average density

⟨ni⟩=G1(xi,θ)Δx.

(3)

The value of θ is determined from the least squares formula

ddθM∑i=1(¯ni−⟨ni⟩)22Δ2ni/m=0.

(4)

The fluctuations in each bin, Δ2ni=limm→∞∑mk=1(n(k)i−⟨ni⟩)2,
are calculated from the probability p(ni|θ) of detecting ni particles in the i-th bin (for details of derrivation, see A),

In Eq. (4), ⟨ni⟩ and Δ2ni are assumed to be known, since in the phase estimation stage the only measured data are ¯ni.
The quantities ⟨ni⟩ and Δ2ni are instead constructed during the calibration stage, preceding the phase estimation stage,
by repeating the experiment with different known values of θ. If the number of experiments in the calibration is large,
and in absence of thermal and technical noise, the measured ⟨ni⟩ and Δ2ni will tend to the theoretical predictions
given in Eq. (3) and (6), respectively.

Our goal at this point is to determine how the quantum fluctuations Δ2ni in the i-th bin influence the sensitivity of the phase estimation via the fit (4).
To this end, we employ the concept of the maximum likelihood estimation (MLE) [27, 28].

Figure 2:
Sensitivity of the phase estimation from the fit to the density,
as a function of |ψin⟩∈A (solid black line) plotted using
√mΔθ with Eq.(11) and N=100 particles.
The blue dashed
line represents the shot-noise limit. The horizontal dotted red line
indicates the position of the coherent state.
Clearly, the sensitivity is bounded by the shot-noise. The inset shows the
behavior of the sensitivity in the vicinity of the coherent state.

If some quantity ξ is measured, the MLE is defined as the choice of θ
which maximizes the conditional probability P(ξ|θ) for the occurrence of ξ given θ. That is, the phase θ
is estimated from the condition ddθP(ξ|θ)=0.
In case of the fit discussed here, the estimation is based on the measured average occupations ¯ni.
If the number of measurements m is large, then according to the central limit theorem the
probability distribution for the average ¯ni in the i-th bin tends to the Gaussian
p(¯ni|θ)=1√2πΔni/√me−(¯ni−⟨ni⟩)22Δ2ni/m.
In every shot the atom counts are correlated between the bins.
However, in order to link the MLE with the sensitivity of the least squares fit, we construct the likelihood function as if the measurement results
¯ni and ¯nj, with i≠j, were uncorrelated.
Thus the total probability of measuring the values
{¯n}=(¯n1…¯nM) is a product
P({¯n}|θ)=∏Mi=1p(¯ni|θ). We note that in this case, indeed, the condition for the MLE,
ddθP({¯n}|θ)=0 coincides with Eq.(4).
The MLE sensitivity saturates the Cramer-Rao Lower Bound [27, 28], Δ2θ=F−1. Here F is the Fisher information (FI),

F=N∑¯n1…¯nM=01P({¯n}|θ)(∂∂θP({¯n}|θ))2→m≫1mM∑i=11Δ2ni(∂⟨ni⟩∂θ)2.

(7)

Therefore, the sensitivity for the least squares fit is given by Eq.(7) as well.

In the following, we demonstrate that this sensitivity is bounded by
the shot-noise. Let us assume for the moment
that the second term in the Eq.(6) – which is proportional to (Δx)2 – can be neglected.
Then, as can be seen from Eq.(6), the particle number distribution is Poissonian. The FI from (7) reads

F=mM∑i=11G1(xi,θ)(∂∂θG1(xi,θ))2Δx≃mN∞∫−∞dx1p1(x|θ)(∂∂θp1(x|θ))2,

(8)

with the one-particle probability

p1(x|θ)=12(|ψa(x,t)|2+|ψb(x,t)|2)+2N⟨^Jx⟩Re[ψ∗a(x,t)ψb(x,t)eiθ].

(9)

We now calculate the Fisher information (8) explicitly.
As the interference pattern is formed after long expansion time,
the mode functions can be written as

ψa/b(x,t)≃eix22~σ2∓ix⋅x0~σ2⋅~ψ(x~σ2),

(10)

where ~σ=√ℏtm, ~ψ is a Fourier transform of the initial wave-packets, common
for ψa and ψb and the separation of the wells is 2x0. This gives

F=mN∫∞−∞dx∣∣~ψ(x~σ2)∣∣2a2sin2φ1+acosφ,

with a=2N⟨^Jx⟩ and φ=2x⋅x0~σ2+θ. Notice that when the
expansion time is long, the function ~ψ varies slowly, as compared to
the period of oscillations of sinφ and cosφ. Therefore, in the above expression, one can substitute the
oscillatory part with its average value. Using the normalization of ~ψ we obtain

F=mNa221−(−1+√1−a2)21+a(−1+√1−a2)

(11)

As 0≤a≤1, we have 0≤F≤mN. Therefore the Fisher information for the fit is always smaller than the shot-noise, giving Δθ≥ΔθSN=1√m1√N for any two-mode input state
(ΔθSN denotes the shot-noise sensitivity).
Below we argue that the inclusion of the second term in the fluctuations in Eq. (6) does not improve the sensitivity.

In Eq.(6), the first term G1(xi,θ)Δx scales linearly with N while
the second term, as a function of N, is a polynomial of the order not higher than two,
aN2+bN+c. The fluctuations Δ2ni must be positive, thus a≥0.
Otherwise, for large N, no matter how small Δx, we would have Δ2ni<0.
If a>0, the positive second term enlarges the fluctuations and thus worsens the sensitivity.
When a=0, the
first and second terms in Eq.(6) scale linearly with N,
and for small Δx the second term can be neglected, thus
we end up again with Eq.(8)111
A similar argument shows that increasing Δx also worsens the sensitivity with respect to Eq.(8).

In order to calculate the sensitivity Δθ=1√F in Eq.(11),
we consider the ground states
of the BEC in a double well potential. In the two-mode approximation, the Hamiltonian of the system reads

^H=−EJ^Jx+ECN^J2z.

(12)

We construct a family of states A by finding ground states of the above Hamiltonian for various values of the ratio
γ=ECNEJ. And so, for γ>0, the elements of A are number-squeezed states and tend to the twin-Fock state
|ψin⟩=∣∣N2,N2⟩ with γ→∞.
For γ<0 the elements of A are phase-squeezed states [23]. With γ→−∞, the ground state of (12) tends to the NOON state
|ψin⟩=1√2(|N0⟩+|0N⟩).
With γ=0 we have a coherent state, |ψin⟩=1√N!(^a†+^b†√2)N|0⟩.
Notice that for all |ψin⟩∈A, the coefficients Cn, which were introduced in previous section, are real and symmetric.
For each state in A, we calculate a=2N⟨^Jx⟩, and insert it into Eq.(11).
The sensitivity shown in Fig.2 is clearly bounded by the shot noise.

This limitation for the sensitivity
can be explained as follows. The value of the
FI given by Eq.(8)
is expressed in terms of the single-particle probability. We expect the useful non-classical many body correlations to decrease the value of Δθ, but
the FI (8) is insensitive to these correlations, and thus must be bounded by the shot-noise. In the next section we demonstrate
that the estimation based on the measurement of position correlations can improve the phase sensitivity.

4 Estimation via the correlation functions

In the estimation protocol discussed in this section, the phase θ is deduced from the measurement of the
k-th order correlation function
Gk(→xk|θ). As previously, we choose to deduce θ using the MLE:
a set of k positions →xk is measured, and the phase is chosen from the condition ddθGk(→xk|θ)=0.
After m≫1 experiments, Δ2θ=F−1(k), where

F(k)=mNk∞∫−∞d→xk1pk(→xk|θ)(∂∂θpk(→xk|θ))2,

(13)

with pk(→xk|θ)=(N−k)!N!Gk(→xk,θ). The coefficient Nk stands for the number
of independent drawings of k particles from N, i.e. (Nk)/(N−1k−1). We notice
that by setting k=1, i.e. the estimator is a single-particle density,
we recover the FI from Eq.(8). Therefore, the
measurement of positions of N particles used as independent is, in terms of sensitivity,
equivalent to fitting the average density to the interference pattern, and is limited by the shot-noise.

Let us calculate the FI for the case k=N, corresponding to the measurement of the full N-body correlation function. We represent the mode functions ψa/b using
Eq.(10).
This expression allows to calculate uθ(x,φ), and, in turn, the probability (2), which is then inserted into
Eq.(13). The integrals over space can be performed analytically (see B for details)
and the outcome is

F(k=N)=m⋅4N∑n=0C2n(n−N2)2=m⋅4Δ2^Jz=FQ.

(14)

Here, by FQ we denote the QFI, which is a maximal value of the Fisher information with respect to all
possible measurements [24]. In the case of pure
states, the value of the QFI is given by 4m times the variance of the phase-shift generator, thus in our case it reads FQ=m⋅4Δ2^Jz.
As denoted by open circles in Fig.3, the Eq.(14)
gives Δθ=√mΔθSN for the coherent state (γ=0), and overcomes
this bound for all |ψin⟩∈A with γ<0. The NOON state
gives the Heisenberg limit, ΔθHL=1√m1N.

Figure 3:
The sensitivity √mΔθ (black solid lines) for N=8
calculated by numerical integration of the Eq.(13) for various k as a function of
|ψin⟩∈A with γ<0. The two limits,
√mΔθSN and √mΔθHL are
denoted by the upper and lower dashed blue lines, respectively. The optimal sensitivity, given by the
QFI, is drawn with red open circles.
The inset magnifies the vicinity of the coherent state, showing that the sub-shot-noise sensitivity is reached
starting from kmin=4.

We now discuss how the sensitivity given by the inverse of Eq.(13) changes for k<N.
The space integrals for k≠1,N cannot be evaluated analytically, thus we calculate the FI numerically taking Gaussian wave-packets

~ψ(x~σ2)=(2σ20π~σ4)14e−x2σ20~σ4

(15)

with the initial width σ0=0.1 and
half of the well separation x0=1.
The Fig.3 shows how the sensitivity from Eq.(13) for N=8 atoms changes with increasing k as a function of |ψin⟩∈A.
The sensitivity improves with growing k, and goes below the shot-noise limit at kmin=4.

For higher numbers of particles, we numerically checked that kmin tends to √N. Therefore, one would have to measure the correlation function of the order of
at least √N in order to beat the shot-noise limit. In a realistic experiment with cold atoms, where N≃1000, it would be very difficult to use the correlation function
of such high order for phase estimation. The biggest difficulty resides in the calibration stage, during which one would need to experimentally probe
a function of a k dimensional variable →xk (with k>√N) for different values of theta.

In the following section we present a phase estimation scheme based on the measurement of the center-of-mass of the interference pattern. Although the probability for measuring
the center-of-mass at position x is a function of just a one-dimensional variable, it can still provide the sub-shot-noise sensitivity.
Nevertheless, we will demonstrate that the implementation of this estimation protocol can be challenging.

5 Estimation via the center-of-mass measurement

5.1 Measurement of all N atoms

In order to estimate θ from the measurement of the center-of-mass, one has to go through a relatively simple
calibration stage.
Positions of N atoms are recorded independently and from this data location of the center-of-mass is deduced.
Many repetitions of the experiment give the function pcm(x|θ) of a one-dimensional variable.
The expression for this function can be extracted from the full N-body probability (2) by

pcm(x|θ)=∫d→xNδ(x−1NN∑i=1xi)pN(→xN|θ),

where “δ” is the Dirac delta.
To provide an analytical expression for this probability, we model the mode-functions by Gaussians as in Eq.(15). Using a reasonable assumption that the initial
separation of the wave-packets is much larger than their width, i.e. e−x20/σ20≪1, we obtain

pcm(x|θ)=√2σ20Nπ~σ4e−2x2σ20~σ4N[1+12(C0+CN)2cos(Nθ+2Nx0~σ2x)].

(16)

The details of this derivation are presented in C.
Notice an interesting property – the above probability
depends on θ only for states with non-negligible NOON components C0 and CN, as already noticed in [29].

When pcm(x|θ) is known,
the phase can be estimated using the MLE, as used in the previous sections. Then once again
the sensitivity is given by the inverse of the FI, which can be calculated analytically,

Fcm=m∞∫−∞dxpcm(x|θ)(∂∂θpcm(x|θ))2=mN2[1−√1−12(C0+CN)2],

(17)

where m is the number of experiments.
In Fig.4 we plot the sensitivity calculated by the inverse of the FI (17) as
a function of |ψin⟩∈A with γ≤0.
Although the estimation through the center-of-mass is not optimal (Δθ>1√FQ), the sensitivity can be better than the shot-noise,
with Δθ→ΔθHL for |ψin⟩→ NOON.
The calibration stage is not as difficult as in the case of high-order correlations, however
phase estimation based on the center-of-mass measurement demands detection of all N atoms [30, 31, 32, 33], as we show below.

Figure 4:
The sensitivity √mΔθ (black solid line) for N=100
calculated with Eq.(17) as a function of |ψin⟩∈A with γ<0.
The values of √mΔθSN and √mΔθHL are
denoted by the upper and lower dashed blue lines, respectively. The optimal sensitivity, given by the inverse of
the QFI, is drawn with the red open circles.

5.2 Measurement of k<N atoms

If the measurement of the center-of-mass is based on detection of k<N atoms, the probability (16) transforms into

p(k)cm(x|θ)=∫d→xkδ(x−1kk∑i=1xi)pk(→xk|θ),

(18)

where pk(→xk|θ)=∫d→xN−kpN(→xN|θ). The probability (18) can be calculated
in a manner similar to that presented in C. The result is

p(k)cm(x|θ)=√2σ20kπ~σ4e−2x2σ20~σ4k[1+acos(kθ+2kx0~σ2x)],

(19)

where

a=2N−k∑i=0((N−k)i)CiCi+k√(Ni+k)(Ni).

(20)

Notice that for k=N we recover the result from the previous section a=2C0CN=12(C0+CN)2
(as we are using the symmetric states, C0=CN). The FI for the probability (19) can be calculated analytically,

F=mk2(1−√1−a2).

(21)

Let us now evaluate a – and thus F – for various k≃N. For k=N and the NOON state, we have
C0=CN=1√2, giving a=1 and F=mN2. From Eq.(20) we notice that, for any k,
a is the sum of N−k terms, each depending on the coefficients Ci and Ci+k. And so, for k=N−1, a will be maximal for a NOON-like state with C0=CN−1=12 and C1=CN=12. For this state we obtain a=1√N, and for large N the value of the FI is F=mN. Therefore, the phase estimation using the center-of-mass of N−1 particles gives a sensitivity bounded by the shot-noise.
Each loss of an atom decreases the FI roughly by a
factor of N, drastically deteriorating the sensitivity.
In Fig.5 we plot the sensitivity √mΔθ calculated with the FI
from Eq.(21) for various k≃N. To calculate a, we choose the subset of |ψin⟩∈A
which are in the vicinity of the NOON state. The Figure shows a dramatic loss of sensitivity as soon as k≠N.

Figure 5:
The sensitivity √mΔθ (black solid line) for N=100
calculated with Eq.(21) as a function of |ψin⟩∈A with γ<0.
The values of √mΔθSN and √mΔθHL are
denoted by respectively the upper and the lower dashed blue line. The three solid lines correspond to the
phase sensitivity for estimation of the center-of-mass with different number of particles. For k=100,
the sensitivity is below the shot-noise and tends to √mΔθHL for |ψin⟩→
NOON. As soon as k≠N, the sub-shot-noise sensitivity is lost and the value of √mΔθ
increases dramatically.

6 Estimation via the position measurement with the Mach-Zehnder Interferometer

6.1 Formulation

So far, we focused on the position
measurement of atoms released from a double-well trap, and studied the phase estimation sensitivity.
The fit to the density gives sensitivity limited by the shot-noise, and
this bound can be overcome by phase estimation with correlation functions of the order of at least √N.
As it is difficult to measure these correlations in the experiment, it will be challenging to beat the shot-noise
limit using the interference pattern.
Although the sensitivity of the phase estimation based on the center-of-mass measurement can also be sub-shot-noise,
the protocol is extremely vulnerable to the loss of particles.

In the above scenario, the sub-shot-noise sensitivity, which relies on non-classical particle correlations,
is reached by directly measuring spatial correlations between the atoms forming the interference pattern and using the
latter as estimators for the phase shift. On the other hand, it is well known that the Mach-Zehdner Interferometer (MZI)
can easily provide sub-shot-noise sensitivity just by a simple measurement of the population imbalance between
the two arms and a proper choice of the input state |ψin⟩.
This is because, in the MZI, the correlations between the two modes carry the part of the information contained
in the particle correlations which is useful for phase estimation. When the clouds are released
from the trap and the two modes start to overlap, the correlations between the two modes are lost,
since an atom detected in the overlap region cannot be told to have come from either of the two initially separated clouds.
This is the reason why it is necessary then to measure directly high-order spatial correlation functions in
order to reach sub-shot-noise sensitivity.

It would be thus interesting to quantify the effect of the wave-packets’ overlap on the sensitivity of the MZI.
This analysis has also a practical interest since, in the implementation of the atomic MZI,
the precision of the population imbalance measurement can be improved by opening the trap and letting
the clouds expand for a while. In this way the density of the clouds drops, facilitating the measurement
of the number of particles. However, during the expansion, the clouds inevitably start to overlap, leading to loss of
information about the origin of the particles, as noted above.
In this section, we show how the increasing overlap deteriorates the sensitivity of the MZI in two estimation scenarios.

The MZI consists of three stages: two beam-splitters represented by unitary evolution operators e∓iπ2^Jx
separated by the phase acquisition e−iθ^Jz.
The atomic MZI can be realized as follows. Consider a two-mode system governed by the Hamiltonian (12) with EC=0.
The first beam-splitter is done by letting the atoms tunnel between the two wells for t=π2ℏEJ so
the unitary evolution operator reads ^U1=e−iπ2^Jx. Then, an inter-well barrier is raised,
in order to supress the oscillations (EJ=0) and a phase between the wells is imprinted, giving
^U2=e−iθ^Jz. The interferometric sequence
is closed by another beam-splitter, ^U3=eiπ2^Jx. The full evolution operator reads

^U(θ)=^U3^U2^U1=eiπ2^Jxe−iθ^Jze−iπ2^Jx=e−iθ^Jy,

where we used commutation relations of the angular momentum operators.

In order to analyze the sensitivity of the MZI,
we introduce the conditional probability pN(→xN|θ) of detecting N atoms at positions
→xN=x1…xN. To evaluate this probability for any initial state of the double-well system
|ψin⟩, we take the same steps as in Section II. In the Heisenberg representation, the field operator
evolves as

^Ψθ(x,t)≡^U†(θ)^Ψ(x,t)^U(θ)

(22)

=[ψa(x,t)cos(θ2)+ψb(x,t)sin(θ2)]^a

+[ψb(x,t)cos(θ2)−ψa(x,t)sin(θ2)]^b.

Again, we express the action of the field operator on |ψin⟩ using the basis of the coherent phase-states
and obtain Eq.(1) with

uθ(x,φ;t)=

=[ψa(x,t)cos(θ2)+ψb(x,t)sin(θ2)]eiφ2

+[ψb(x,t)cos(θ2)−ψa(x,t)sin(θ2)]e−iφ2.

Therefore, the probability pN(→xN|θ) for the MZI is given by Eq.(2) with the
uθ(x,φ;t) function defined above.

6.2 Measurement of the population imbalance

The most common phase-estimation protocol discussed in context of the MZI is the measurement of the population imbalance
between the two arms of the interferometer. In order to assess how the sensitivity of this protocol is influenced
by the expansion of the wave-packets, we introduce the probability of measuring nL atoms in the left sub-space as follows

pimb(nL|θ)=(NnL)0∫−∞d→xnL∞∫0d→xN−nLpN(→xN|θ).

(23)

This probability depends on the expansion time via ψa,b(x,t), which enter the definition of
pN(→xN|θ).
The sensitivity for various expansion times, if m≫1 measurements are performed, can be calculated using the error propagation formula,

Δ2θ=1mΔ2n∣∣∂⟨n⟩∂θ∣∣2,

(24)

where

⟨n⟩=N∑nL=0pimb(nL|θ)(nL−N2)

is the average value of the population imbalance and

Δ2n=N∑nL=0pimb(nL|θ)(nL−N2)2−⟨n⟩2

are the associated fluctuations. The probability (23) resembles
Eq.(5), and the above moments are calculated as in A resulting in

⟨n⟩=∞∫0dxG1(x|θ)−N2andΔ2n=N24−∞∫00∫−∞d→x2G2(→x2|θ)−⟨n⟩2.

The two lowest correlation functions for the MZI read
G1(x|θ)=⟨^Ψ†θ(x,t)^Ψθ(x,t)⟩ and
G2(→x2|θ)=⟨^Ψ†θ(x1,t)Ψ†θ(x2,t)^Ψθ(x2,t)^Ψθ(x1,t)⟩, with
the field operator from Eq.(22), and the averages calculated with the input state. When the
two wave-packets don’t overlap, i.e. ψa(x,t)ψ∗b(x,t)≃0 for all x∈R,
Eq.(24) simplifies to

Δ2θ=1mΔ2^Jxsin2θ+⟨^J2z⟩cos2θ⟨^Jx⟩2cos2θ.

(25)

This is the well-known expression for the sensitivity of the population imbalance between separated arms.
It gives Δθ≤ΔθSN for all |ψin⟩∈A with γ≥0.

We investigate the impact of the overlap on the sensitivity (24) by modelling the free expansion
of the wave-packets ψa/b(x,t) by Gaussians,

ψa/b(x,t)=1(2πσ20(1+iτ))1/4e−(x±x0)24σ2(1+iτ),

and take x0=1 and the initial width σ0=0.1. In Fig. 6
we plot the sensitivity √mΔθ taking θ=0 and N=100 for three different expansion times
τ. The initial sensitivity deteriorates as soon as the condensates start to overlap, and the sub-shot-noise
sensitivity is lost for long expansion times. We attribute this decline to the loss of information
about the correlations between the modes. Therefore, special attention has to be paid to avoid the overlap
when letting the two trapped
condensates spread. Although we expect that the expansion facilitates the atom-number measurement, the conclusion of this
section is that any overlap of the spatial modes has a strong negative impact on the sensitivity of the MZI.

Figure 6:
(a) The sensitivity √mΔθ calculated with Eq.(24) for
three different expansion times τ as a function of |ψin⟩∈A with γ≥0.
The solid black line corresponds to the setup shown in (b), where τ=0 and the wave-packets don’t overlap.
The dashed red line corresponds to (c), where τ=3 and the wave-packets start to overlap.
The dot-dashed green line corresponds to (d), where τ=10 and the wave-packets strongly overlap.
Clearly, the sensitivity is influenced by any non-vanishing overlap.
The values of √mΔθSN and √mΔθHL are
denoted by respectively the upper and the lower dashed blue line. Here, N=100 and θ=0.

6.3 Estimation via the center-of-mass measurement for the MZI

In Section 5.1, we have demonstrated that when the two wave-packets overlap and form an interference pattern, phase estimation based on the
center-of-mass measurement can give sub-shot-noise
sensitivity. Here we study the same estimation strategy applied in the MZI case.
Again, we start with the probability
pcm(x|θ) of measuring the center-of-mass at position x,

pcm(x|θ)=∫d→xNδ(x−1NN∑i=1xi)pN(→xN|θ).

Using pN(→xN|θ) for the MZI, one can analytically
calculate the center-of-mass probability only in the limit of small θ,

pcm(x|θ)=N2πσ2[N∑l=0C2lfl(x)+θN∑l=0ClCl+1√(l+1)(N−l)(fl+2(x)−fl(x))],

(26)

where

fl(x)=exp⎡⎢
⎢
⎢⎣−(x−x0(2lN−1))22σ20⎤⎥
⎥
⎥⎦.

With this probability, we can again calculate the sensitivity using the error propagation formula [27, 28],

Δ2θ=1mΔ2x∣∣∂⟨x⟩∂θ∣∣2,

where

⟨x⟩=∞∫−∞dxpcm(x|θ)xandΔ2x=∞∫−∞dxpcm(x|θ)x2−⟨x⟩2.

These two moments can be easily calculated with Eq.(26), giving, in the limit θ→0,

Δ2θ∣∣θ→0=1m⎡⎣⟨^J2z⟩⟨^Jx⟩2+(σx0)2N4⟨^Jx⟩2⎤⎦.

(27)

Notice that when the initial size of the Gaussians tends to zero, we recover the sensitivity from Eq.(25)
(in the limit of θ→0). This is not surprising, as when the mode-function are point-like, the
measurements of the center-of-mass and the measurement of the population imbalance are equivalent, and related
by xcm=2x0nN. Therefore, for small σ, the measurement of the center-of-mass yields sub-shot-noise
sensitivity for all |ψin⟩∈A with γ>0. However, for non-zero σ, the second term in
Eq.(27) spoils the sensitivity. This is because N4⟨^Jx⟩2≥1N
is always satisfied.
Even if the first term scales better than at the shot-noise limit, the other one does not,
and will dominate for large N.

¿From what we presented in this Section, we conclude that both the population imbalance and the center-of-mass measurements can give sub-shot-noise sensitivity
for the MZI, but both are very sensitive to the growing size of the wave-packets.

7 Conclusions

In this manuscript we have discussed in detail how the measurement of positions
of atoms forming an interference pattern can be useful in context of atom interferometry.
We showed that the phase estimation based on the fit to the density gives sensitivity limited by the shot-noise, because the FI is expressed in terms
of the single-particle probability only. The sensitivity can be improved below the shot-noise limit by estimating the phase using correlation functions of order at least √N.
Moreover, we demonstrated that the information contained in the N-th order correlation function allows to perform an optimal detection strategy,
reaching Heisenberg-limited sensitivity when NOON states are used.
We also showed that the measurement of the position of the center-of-mass of the interference pattern
gives sub-shot-noise sensitivity for all states with non-negligible NOON component.
Both the measurement of high-order correlations and the center-of-mass position
are difficult to perform. The former requires the construction of a function of highly-dimensional variables,
while the latter works well only if all N atoms forming the interference pattern are detected.
We attribute the difficulty to obtain the sub-shot-noise sensitivity to the fact
that, after formation of the interference pattern, the modes cannot be distinguished,
and the information useful for interferometry is only contained in the correlations between the particles.
These correlations are very difficult to extract from the experimental data,
therefore reaching sub-shot-noise sensitivity with two interfering BECs might prove very challenging.
In the final part of this work, we turned our attention to the MZI, which is known to provide sub-shot-noise sensitivity for the simpler measurement of the
population imbalance between the two clouds. We have shown that the sensitivity of the MZI is strongly influenced by a non-zero overlap between the two wave-packets,
both in case of the population imbalance and the center-of-mass measurement.

Appendix A Bin fluctuations

To derive the expression for the average number and the fluctuations of the atom count in a bin, we use the
probability from Eq.(5). With help of Eq.(2)
we obtain